# Elon Lindenstrauss (The Hebrew University of Jerusalem)

**Lundi 4 juin 2012, 16h00 / Monday, June 4, 2012, 4:00 pm**

**Salle – Room 6214**

Centre de recherches mathématiques, Pavillon André-Aisenstadt,

Université de Montréal, 2920, chemin de la Tour

Cette conférence s'adresse à un large auditoire / Suitable for a general audience

**Entropy and Quantum Unique Ergodicity**

The relationship between classical dynamics of systems and their quantum behavior produces a rich trove of interesting and deep questions. One of these questions is the study of asymptotic distribution of eigenfunctions of the Laplacian on a compact (or noncompact) manifold in the high eigenvalue (i.e. semiclassical) limit. Such eigenfunctions correspond to the steady states of the quantum dynamics of a single particle constraints to lie on the manifold with no external force applied; the corresponding classical dynamics is the geodesic flow on the manifold.

The Quantum Unique Ergodicity Conjecture of Rudnick and Sarnak states that if the manifold has negative sectional curvature (which implies that the classical dynamics is uniformly hyperbolic, hence "chaotic"), the eigenfunctions of the Laplacian should become equidistributed in the semiclassical limit. Recently there has been progress towards this conjecture, a substantial portion of which hinges upon the notion of Kolmogorov-Sinai (a.k.a. ergodic theoretic) entropy.

In my talk I will survey some of these results; no prior knowledge in ergodic theory or spectral theory will be assumed.

### Conférences dans le cadre de l'atelier sur la géométrie des valeurs propres et fonctions propres (4-8 juin 2012)

Lectures at the Workshop on Geometry of Eigenvalues and Eigenfunctions (June 4-8, 2012)

**Mardi 5 juin 2012, 16h00 / Tuesday, June 5, 2012, 4:00 pm
Jeudi 7 juin 2012, 16h00 / Thursday, June 7, 2012, 4:00 pm**

**Salle – Room 6214
**Centre de recherches mathématiques, Pavillon André-Aisenstadt,

Université de Montréal, 2920, chemin de la Tour

## Arithmetic Quantum Unique Ergodicity I & II

In these two talks (which will be independent from my first talk) I would discuss a special instance of the quantum unique ergodicity question, where the manifold is a finite area arithmetic surface.

Such surfaces have a lot of symmetry, albeit of a slightly subtle nature, provided by the Hecke operators. The Laplacian eigenfunctions which respect all of these symmetries are called Hecke-Maass forms and play a surprisingly important role in modern analytic number theory. Thanks to this extra symmetry, much more is known regarding the properties of such eigenfunctions than of general eigenfunctions.

I will survey recent results that were obtained regarding these eigenfunctions (and the closely related class of holomorphic forms) using both number theoretic and dynamical techniques by several researchers.

I will present in more detail my recent joint work with S. Brooks which relates the study of eigenfunctions (or even quasimodes) on arithmetic surfaces to the study of eigenfunctions of the discrete Laplacian on finite graphs, and provides arithmetic quantum unique ergodicity for quasimodes of the Laplacian and one Hecke operator which in certain respects is optimal.